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      第三周矩阵的四则运算
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        <h1 id="矩阵运算及应用"><a href="#矩阵运算及应用" class="headerlink" title="矩阵运算及应用"></a>矩阵运算及应用</h1><h2 id="矩阵代数理论"><a href="#矩阵代数理论" class="headerlink" title="矩阵代数理论"></a>矩阵代数理论</h2><p>多个线性系统相互联结，构建更大，更复杂的系统，建立矩阵代数理论</p>
<h2 id="矩阵的加法"><a href="#矩阵的加法" class="headerlink" title="矩阵的加法"></a>矩阵的加法</h2><h3 id="超市上，下半年营业额"><a href="#超市上，下半年营业额" class="headerlink" title="超市上，下半年营业额"></a>超市上，下半年营业额</h3><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A = randi([<span class="number">1</span> <span class="number">100</span>],<span class="number">5</span>,<span class="number">5</span>)</span><br><span class="line">A =</span><br><span class="line">      <span class="number">64</span>             <span class="number">54</span>             <span class="number">88</span>             <span class="number">42</span>             <span class="number">15</span></span><br><span class="line">      <span class="number">29</span>             <span class="number">45</span>             <span class="number">28</span>             <span class="number">21</span>             <span class="number">17</span></span><br><span class="line">      <span class="number">54</span>             <span class="number">13</span>             <span class="number">21</span>             <span class="number">95</span>             <span class="number">63</span></span><br><span class="line">      <span class="number">70</span>             <span class="number">50</span>             <span class="number">57</span>              <span class="number">9</span>             <span class="number">58</span></span><br><span class="line">      <span class="number">50</span>             <span class="number">86</span>             <span class="number">65</span>             <span class="number">11</span>              <span class="number">6</span></span><br><span class="line">&gt;&gt; B = randi([<span class="number">1</span> <span class="number">100</span>],<span class="number">5</span>,<span class="number">5</span>)</span><br><span class="line">B =</span><br><span class="line">      <span class="number">94</span>             <span class="number">94</span>             <span class="number">18</span>             <span class="number">31</span>              <span class="number">3</span></span><br><span class="line">      <span class="number">73</span>             <span class="number">99</span>             <span class="number">40</span>             <span class="number">30</span>             <span class="number">85</span></span><br><span class="line">      <span class="number">74</span>             <span class="number">86</span>             <span class="number">14</span>             <span class="number">34</span>             <span class="number">56</span></span><br><span class="line">       <span class="number">7</span>             <span class="number">79</span>              <span class="number">4</span>             <span class="number">47</span>             <span class="number">86</span></span><br><span class="line">      <span class="number">87</span>             <span class="number">52</span>             <span class="number">94</span>             <span class="number">65</span>             <span class="number">35</span></span><br><span class="line">&gt;&gt; A+B</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">     <span class="number">158</span>            <span class="number">148</span>            <span class="number">106</span>             <span class="number">73</span>             <span class="number">18</span></span><br><span class="line">     <span class="number">102</span>            <span class="number">144</span>             <span class="number">68</span>             <span class="number">51</span>            <span class="number">102</span></span><br><span class="line">     <span class="number">128</span>             <span class="number">99</span>             <span class="number">35</span>            <span class="number">129</span>            <span class="number">119</span></span><br><span class="line">      <span class="number">77</span>            <span class="number">129</span>             <span class="number">61</span>             <span class="number">56</span>            <span class="number">144</span></span><br><span class="line">     <span class="number">137</span>            <span class="number">138</span>            <span class="number">159</span>             <span class="number">76</span>             <span class="number">41</span></span><br></pre></td></tr></table></figure>
<h2 id="矩阵的数乘"><a href="#矩阵的数乘" class="headerlink" title="矩阵的数乘"></a>矩阵的数乘</h2><h3 id="平时成绩和期末成绩占比"><a href="#平时成绩和期末成绩占比" class="headerlink" title="平时成绩和期末成绩占比"></a>平时成绩和期末成绩占比</h3><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; <span class="number">0.3</span>*A+<span class="number">0.7</span>*B</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">      <span class="number">85</span>             <span class="number">82</span>             <span class="number">39</span>            <span class="number">343</span>/<span class="number">10</span>          <span class="number">33</span>/<span class="number">5</span></span><br><span class="line">     <span class="number">299</span>/<span class="number">5</span>          <span class="number">414</span>/<span class="number">5</span>          <span class="number">182</span>/<span class="number">5</span>          <span class="number">273</span>/<span class="number">10</span>         <span class="number">323</span>/<span class="number">5</span></span><br><span class="line">      <span class="number">68</span>            <span class="number">641</span>/<span class="number">10</span>         <span class="number">161</span>/<span class="number">10</span>         <span class="number">523</span>/<span class="number">10</span>         <span class="number">581</span>/<span class="number">10</span></span><br><span class="line">     <span class="number">259</span>/<span class="number">10</span>         <span class="number">703</span>/<span class="number">10</span>         <span class="number">199</span>/<span class="number">10</span>         <span class="number">178</span>/<span class="number">5</span>          <span class="number">388</span>/<span class="number">5</span></span><br><span class="line">     <span class="number">759</span>/<span class="number">10</span>         <span class="number">311</span>/<span class="number">5</span>          <span class="number">853</span>/<span class="number">10</span>         <span class="number">244</span>/<span class="number">5</span>          <span class="number">263</span>/<span class="number">10</span></span><br><span class="line">&gt;&gt; A = <span class="built_in">ones</span>(<span class="number">5</span>)</span><br><span class="line">A =</span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">       <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span>              <span class="number">1</span></span><br><span class="line">&gt;&gt; <span class="number">5</span>*A</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span>              <span class="number">5</span></span><br><span class="line"> &gt;&gt; A+A</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line">       <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span>              <span class="number">2</span></span><br><span class="line"></span><br><span class="line">&gt;&gt; A+A+<span class="number">4</span>*A</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span>              <span class="number">6</span></span><br><span class="line">&gt;&gt;</span><br></pre></td></tr></table></figure>
<h2 id="矩阵的线性运算"><a href="#矩阵的线性运算" class="headerlink" title="矩阵的线性运算"></a>矩阵的线性运算</h2><div class="table-container">
<table>
<thead>
<tr>
<th>规则</th>
<th>表达式</th>
</tr>
</thead>
<tbody>
<tr>
<td>交换律</td>
<td>A+B = B+A</td>
</tr>
<tr>
<td>结合律</td>
<td>A+(B+C) = (A+B) +C</td>
</tr>
<tr>
<td>数乘结合律</td>
<td>($\lambda <em> \mu$) A = $\lambda$ </em> ( $\mu$ A)</td>
</tr>
<tr>
<td>数乘分配律</td>
<td>$\lambda$ (A+B)  = $\lambda$ A + $\lambda$ B</td>
</tr>
</tbody>
</table>
</div>
<h1 id="矩阵的乘法"><a href="#矩阵的乘法" class="headerlink" title="矩阵的乘法"></a>矩阵的乘法</h1><h2 id="多服装厂协同生产"><a href="#多服装厂协同生产" class="headerlink" title="多服装厂协同生产"></a>多服装厂协同生产</h2><div class="table-container">
<table>
<thead>
<tr>
<th></th>
<th></th>
<th></th>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td>单位产量/厂名</td>
<td>甲</td>
<td>乙</td>
<td>丙</td>
<td>丁</td>
</tr>
<tr>
<td>帽子</td>
<td>20</td>
<td>4</td>
<td>2</td>
<td>7</td>
</tr>
<tr>
<td>上衣</td>
<td>10</td>
<td>18</td>
<td>5</td>
<td>6</td>
</tr>
<tr>
<td>裤子</td>
<td>5</td>
<td>7</td>
<td>16</td>
<td>3</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>生产时长</td>
<td>8</td>
<td>10</td>
<td>5</td>
<td>9</td>
</tr>
</tbody>
</table>
</div>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A</span><br><span class="line">A =</span><br><span class="line">      <span class="number">20</span>              <span class="number">4</span>              <span class="number">2</span>              <span class="number">7</span></span><br><span class="line">      <span class="number">10</span>             <span class="number">18</span>              <span class="number">5</span>              <span class="number">6</span></span><br><span class="line">       <span class="number">5</span>              <span class="number">7</span>             <span class="number">16</span>              <span class="number">3</span></span><br><span class="line">&gt;&gt; b</span><br><span class="line">b =</span><br><span class="line">       <span class="number">8</span></span><br><span class="line">      <span class="number">10</span></span><br><span class="line">       <span class="number">5</span></span><br><span class="line">       <span class="number">9</span></span><br><span class="line">&gt;&gt; A*b</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">     <span class="number">273</span></span><br><span class="line">     <span class="number">339</span></span><br><span class="line">     <span class="number">217</span></span><br></pre></td></tr></table></figure>
<h2 id="矩阵乘积-定义"><a href="#矩阵乘积-定义" class="headerlink" title="矩阵乘积 定义"></a>矩阵乘积 定义</h2><p>设A 是mxs 矩阵 ，B是 sxn 矩阵，则 AxB 为 mxn的矩阵C， C的各个元素为：</p>
<script type="math/tex; mode=display">
C_{ij} =  \sum_{k=1}^s a_{ik}b_{kj} = a_{i1}*b_{1j} +a_{i2}*b_{j2}+ \cdots+a_{is}*b_{sj}
(i = 1,2,\cdots, m) (j = 1,2, \cdots, n)</script><p>记作：</p>
<script type="math/tex; mode=display">
C = A B</script><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">可乘条件： A的列数 = B的行数</span><br><span class="line">AB乘积的形状： A行 ， B列</span><br><span class="line">AB乘积元素的构成： A的行与B的列的内积</span><br></pre></td></tr></table></figure>
<h2 id="线性变换"><a href="#线性变换" class="headerlink" title="线性变换"></a>线性变换</h2><script type="math/tex; mode=display">
变量 Y = [y_1, y_2 , \cdots, y_m],能够由X = [x_1, x_2, \cdots, x_n] 线性表示，
则成为此变量X到变量Y的线性变换</script><p>多次线性变换等价于矩阵相乘</p>
<p>若 Y = AX ， X = BT，则 Y= ABT</p>
<p>方程组的矩阵表示</p>
<p>A = [ 1, 2, 3]    B = [4; 5; 6]      AB  $ \neq $ BA</p>
<figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br></pre></td><td class="code"><pre><span class="line">&gt;&gt; A = [ <span class="number">1</span>, <span class="number">2</span>, <span class="number">3</span>]</span><br><span class="line">A =</span><br><span class="line">       <span class="number">1</span>              <span class="number">2</span>              <span class="number">3</span></span><br><span class="line">&gt;&gt; B = [<span class="number">4</span>; <span class="number">5</span>; <span class="number">6</span>]</span><br><span class="line">B =</span><br><span class="line">       <span class="number">4</span></span><br><span class="line">       <span class="number">5</span></span><br><span class="line">       <span class="number">6</span></span><br><span class="line">&gt;&gt; A*B</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">      <span class="number">32</span></span><br><span class="line">&gt;&gt;</span><br><span class="line">&gt;&gt;</span><br><span class="line">&gt;&gt; B*A</span><br><span class="line"><span class="built_in">ans</span> =</span><br><span class="line">       <span class="number">4</span>              <span class="number">8</span>             <span class="number">12</span></span><br><span class="line">       <span class="number">5</span>             <span class="number">10</span>             <span class="number">15</span></span><br><span class="line">       <span class="number">6</span>             <span class="number">12</span>             <span class="number">18</span></span><br></pre></td></tr></table></figure>
<p>空间次序不能交换</p>
<ol>
<li>AB C = A （BC）</li>
<li>A （B+C） = AB + AC<br>3.（A + B）C = AC + BC</li>
</ol>
<script type="math/tex; mode=display">A_{mn}I_{n} = I_mA_{mn}</script><p>5.A，B均为上（下）三角矩阵，则C = AB 也为上（下）三角矩阵</p>
<h1 id="矩阵的逆（互为倒数）"><a href="#矩阵的逆（互为倒数）" class="headerlink" title="矩阵的逆（互为倒数）"></a>矩阵的逆（互为倒数）</h1><p>引例：</p>
<p>xy = 1， 则x，y互为倒数</p>
<p>若 X Y = I 则 X Y 互逆</p>
<p>Y = AX 转换成  X = A’ Y</p>
<ol>
<li>A ， $\ A^{-1}$ 逆矩阵唯一</li>
<li>若A B 同阶可逆方阵，满足AB = I ，则BA = I  （AB互逆）</li>
<li>若A可逆，则 $\ A^{-1}$ 也可逆，$\ ( A^{-1})^{-1}$ = A</li>
<li>若A可逆, $\lambda \neq 0$ ,则 $\lambda$ A 可逆</li>
<li>$(AB)^{-1} = B^{-1} A^{-1}$</li>
</ol>
<h2 id="逆矩阵看做-矩阵除法"><a href="#逆矩阵看做-矩阵除法" class="headerlink" title="逆矩阵看做 矩阵除法"></a>逆矩阵看做 矩阵除法</h2><p>若</p>
<script type="math/tex; mode=display">AB = C</script><p>则</p>
<script type="math/tex; mode=display">B = A^{-1}C</script><script type="math/tex; mode=display">A = CB^{-1}</script>
      
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              <ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#矩阵运算及应用"><span class="toc-number">1.</span> <span class="toc-text">矩阵运算及应用</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#矩阵代数理论"><span class="toc-number">1.1.</span> <span class="toc-text">矩阵代数理论</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#矩阵的加法"><span class="toc-number">1.2.</span> <span class="toc-text">矩阵的加法</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#超市上，下半年营业额"><span class="toc-number">1.2.1.</span> <span class="toc-text">超市上，下半年营业额</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#矩阵的数乘"><span class="toc-number">1.3.</span> <span class="toc-text">矩阵的数乘</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#平时成绩和期末成绩占比"><span class="toc-number">1.3.1.</span> <span class="toc-text">平时成绩和期末成绩占比</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#矩阵的线性运算"><span class="toc-number">1.4.</span> <span class="toc-text">矩阵的线性运算</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#矩阵的乘法"><span class="toc-number">2.</span> <span class="toc-text">矩阵的乘法</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#多服装厂协同生产"><span class="toc-number">2.1.</span> <span class="toc-text">多服装厂协同生产</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#矩阵乘积-定义"><span class="toc-number">2.2.</span> <span class="toc-text">矩阵乘积 定义</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#线性变换"><span class="toc-number">2.3.</span> <span class="toc-text">线性变换</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#矩阵的逆（互为倒数）"><span class="toc-number">3.</span> <span class="toc-text">矩阵的逆（互为倒数）</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#逆矩阵看做-矩阵除法"><span class="toc-number">3.1.</span> <span class="toc-text">逆矩阵看做 矩阵除法</span></a></li></ol></li></ol>
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